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11 Mar 2015, 04:56
Kudos
Bookmarks
A
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
65%
(hard)
Question Stats:
68% (03:35) correct
32%
(03:34)
wrong
based on 266
sessions
History
Date
Time
Result
Not Attempted Yet
Attachment:
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A. 1
B. \(\frac{\pi}{2}\)
C. \(\frac{\pi}{3}\)
D. \(\frac{3\pi}{8}\)
E. \(\frac{4\pi}{9}\)
Kudos for a correct solution.
11 Mar 2015, 05:58
Kudos
Bookmarks
Hi
A very time consuming
have edited the pic for all the forum members
Answer in the pic attached
cheers..!!
A very time consuming
have edited the pic for all the forum members
Answer in the pic attached
cheers..!!
Attachments
lune.png [ 28.6 KiB | Viewed 9928 times ]
General Discussion
15 Mar 2015, 21:23
Kudos
Bookmarks
Bunuel
MAGOOSH OFFICIAL SOLUTION:
This problem is based on a famous theorem of Hippocrates of Chios (c. 470 – c. 410 BCE), a predecessor of Euclid. The easiest way to see this is as follows.
Let AC = BC = 2S. This is the radius of the larger circle, which has an area of 4pi*S^2, so that the quarter circle:
must have an area of pi*S^2. Hold that thought.
Now, notice that triangle ABC is a 45-45-90 triangle, so that the length of the hypotenuse must be
\(AB=2\sqrt{2}S\)
and the radius of the smaller circle is
\(AM=\sqrt{2}S\)
The area of that whole circle would be 2pi*S^2, and the area of the semicircle:
would be half of that, pi*S^2. Thus, the quarter circle of the larger circle and semicircle of the smaller circle have the same area. That’s a very big idea!
Now, look at the circular segment:
We don’t need to know that area: simply call it J.
If we subtract the circular segment, J, from the quarter circle of the bigger circle, we get the triangle:
pi*S^2 – J = area of triangle ABC
If we subtract the circular segment, J, from the semicircle of the smaller circle, we get the lune:
pi*S^2 – J = area of the lune
Because those two areas equal the same thing, they must equal each other.
area of triangle ABC = area of the lune
The two are equal, so their ratio is 1.
Answer = (A)
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